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21    PRODUCT OF MEASURES        233

To each real-valued function /on R we associate the function /0 on R2 defined by
MX, y) =/(*) if y < xtf0(xt y) = /0>) if y *> x. Show that/is v-integrable if and only
if /o is (A (x) /z)-integrable, and that in this case \fdv = JT/0 d\ dp,. (Prove the result
first for characteristic functions of intervals.) Hence deduce the formula

jf(x) dw(x) = J/(xXje-) du(x) + J/(*Mjc+) dv(x).

In particular, if u and v are continuous on R, we have the formula of integration
by parts:

Ŗn(x) dv(x) = u(b)v(b) - u(a)v(a) - JV*) du(x).

Consider the case where u and v are constant on each interval [Ģ, n + 1 [ for each
integer Ģ ^ 0 ("Abel's partial summation formula").

7. Let;? be a finite real number j>l, let X and Y be locally compact spaces, A a positive
measure on X and p a positive measure on Y. Let /^ 0 be a function on X x Y
such that/ and /p are (A (x) /j,)-integrable. Prove that

(For each xeX, apply Holder's inequality (13.11, Problem 12(a)) to the function
y*~+f(x, y), in the form/(x, y) = g(x, y)(\ fp(x, y) d\(x)jP\ where q is the exponent
conjugate to p.)

8.   Let X/ (1 <:i<^n) be locally compact spaces and ^ a positive measure on X, for
1 <zi<:n. For each /, let E, denote the product f"] X, . Let// be a function >0 which

n                                  n

is measurable with respect to u = Ž a* on X = 0 X( , and does not depend on xt . If

iĢi                1*1

fl~l is integrable with respect to the measure Ž pj, for 1 <] A: ^w, show that the

function O/' *s /^-integrable and that

•''   /i/2 '' 'fn dpi dfj,2 - - • dp,n < I lp[ J

where Jfc = I • • •   /jj"1 dpv - • • d(j,k-t djjLk+ i • • • t//>tn for I ^k^n. (By induction on

Ģ, using the Lebesgue-Fubini theorem and Holder's inequality.)

Deduce that if A is a /x-measurable subset of X, and At its projection on Et, and
if At is integrable and of measure mi with respect to the measure (x) /z, on Et, then A

is /x-integrable and /x(A) ^ (mi • • • mn)1/(l'1"1>.

Generalize to the case in which, instead of considering the products of the Xi
n — 1 at a time, we consider the (J) products of the X< p at a time, and integrate over X
a product of (J) functions ^0, each of which depends on only p of the variablestion belonging to JT(Y) is the uniform limit of functions belonging